Quantization dimension for infinite self-similar probabilities

The quantization dimension function for a probability measure induced by a set of infinite contractive similarity mappings and a given probability vector is determined. A relationship between the quantization dimension function and the temperature function of the thermodynamic formalism arising in multifractal analysis is established. The result in this paper is an infinite extension of Graf and Luschgy [S. Graf, H. Luschgy, The quantization dimension of self-similar probabilities, Math. Nachr. 241 (2002) 103-109].     

Quantization dimension function and Gibbs measure associated with Moran set

In this paper we consider the Gibbs measure on the one-sided shift dynamical system and determine the quantization dimension function for the image measure supported on a Moran set. A relationship between the quantization dimension function and the temperature function of the thermodynamic formalism arising in multifractal analysis is also established.     

Quantization dimension and temperature function for recurrent self-similar measures

We determine the quantization dimension function for the image measure supported on a recurrent self-similar set of an ergodic Markov measure, and its relationship with the temperature function of the thermodynamic formalism arising in multifractal analysis is established.     

Quantization dimension of random self-similar measures

In this paper, we study the quantization dimension of a random self-similar measure μ supported on the random self-similar set K(ω). We establish a relationship between the quantization dimension of μ and its distribution. At last we give a simple example to show that how to use the formula of the quantization dimension.     

The Quantization of the Cantor Distribution

For a real-valued random variable whose distribution is the classical Cantor probability, the n - quantization error and the n - optimal quantization rules are calculated for every natural number n. Moreover, the connection between the rate of convergence of the logarithms of the quantization errors for n going to infinity and the Hausdorff dimension of the Cantor set is indicated.     

Asymptotics of the quantization errors for in-homogeneous self-similar measures supported on self-similar sets

We study the quantization for in-homogeneous self-similar measures μ supported on self-similar sets.Assuming the open set condition for the corresponding iterated function system, we prove the existence of the quantization dimension for μ of order r ∈(0, ∞) and determine its exact value ξ_r. Furthermore, we show that,the ξ_r-dimensional lower quantization coefficient for μ is always positive and the upper one can be infinite. A sufficient condition is given to ensure the finiteness of the upper quantization coefficient.     

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantization dimension of probability measures supported on Cantor-like sets

Let μ be an arbitrary probability measure supported on a Cantor-like set E with bounded distortion. We establish a relationship between the quantization dimension of μ and its mass distribution on cylinder sets under a hereditary condition. As an application, we determine the quantization dimensions of probability measures supported on E which have explicit mass distributions on cylinder sets provided that the hereditary condition is satisfied.     

A note on the quantization for probability measures with respect to the geometric mean error

We study the quantization with respect to the geometric mean error for probability measures μ on for which there exist some constants C, η > 0 such that for all ε > 0 and all . For such measures μ, we prove that the upper quantization dimension of μ is bounded from above by its upper packing dimension and the lower one is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpiński carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.     

Asymptotic quantization of probability distributions

We give a brief introduction to results on the asymptotics of quantizatlon errors. The topics discussed in-clude the quantization dimension, asymptotic distributions of sets of prototypes, asymptotically optimalquantizations, approximations and random quantizations.     

Main axiom of thermodynamics and entropy of number theory: Tunnel and ultrasecond quantization

We single out the main features of the mathematical theory of equilibrium thermodynamics. The theory of Bose condensate is expressed as a problem in number theory and its relation to various evolutionary processes is studied. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of imperfect gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. We consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and third quantization.     

Optimal Quantization for Uniform Distributions on Cantor-Like Sets

In this paper, the problem of optimal quantization is solved for uniform distributions on some higher dimensional, not necessarily self-similar N-adic Cantor-like sets. The optimal codebooks are determined and the optimal quantization error is calculated. The existence of the quantization dimension is characterized and it is shown that the quantization coefficient does not exist. The special case of self-similarity is also discussed. The conditions imposed are a separation property of the distribution and strict monotonicity of the first N quantization error differences. Criteria for these conditions are proved and as special examples modified versions of classical fractal distributions are discussed. This work contains and generalizes some parts of the authors doctoral thesis (cf. 16).     

On the Range of the Quantization Dimension of Probability Measures on a Metric Compactum

Siberian Mathematical Journal - The quantization dimension of a probability measure on a metric compactum  $ X $ does not exceed the box dimension of the support of the measure....     

Some remarks on absolute continuity and quantization of probability measures

We introduce a notion of monotonicity of dimensions of measures. We show that the upper and lower quantization dimensions are not monotone. We give sufficient conditions in terms of so-called vanishing rates such that νμ implies . As an application, we determine the quantization dimension of a class of measures which are absolutely continuous w.r.t. some self-similar measure, with the corresponding Radon–Nikodym derivative bounded or unbounded. We study the set of quantization dimensions of measures which are absolutely continuous w.r.t. a given probability measure μ. We prove that the infimum on this set coincides with the lower packing dimension of μ. Furthermore, this infimum can be attained provided that the upper and lower packing dimensions of μ are equal.     

Optimal quantization for dyadic homogeneous Cantor distributions

For a large class of dyadic homogeneous Cantor distributions in ?, which are not necessarily self‐similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non‐existence of the quantization coefficient. The class contains all self‐similar dyadic Cantor distributions, with contraction factor less than or equal to 1/3. For these distributions we calculate the quantization errors explicitly. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dimension sets for infinite IFSs: the Texan Conjecture

We consider the set of Hausdorff dimensions of limit sets of finite subsystems of an infinite conformal iterated function system and refer to it as the restricted dimension set. The corresponding set for all subsystems will be referred to as the complete dimension set. We give sufficient conditions for a point to belong to the complete dimension set and consequently to be an accumulation point of the restricted dimension set. We also give sufficient conditions on the system for both sets to be nowhere dense in some interval. Both general results are illustrated by examples. Applying the first result to the case of continued fraction we are able to prove the Texan Conjecture, that is we show that the set of Hausdorff dimensions of bounded type continued fraction sets is dense in the unit interval.     

Badly approximablep-adic integers

It is known that thep-adic integers that are badly approximable by rationals form a null set with respect to Haar measure. We define a [0,1]-valued dimension function on thep-adic integers analogous to Hausdorff dimension inR and show that with respect to this function the dimension of the set of badly approximablep-adic integers is 1.     

Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error.     

The High Resolution Vector Quantization Problem with Orlicz Norm Distortion

We derive a high-resolution formula for the quantization problem under Orlicz norm distortion. In this setting, the optimal point density solves a variational problem which comprises a function g:ℝ₊→[0,∞) characterizing the quantization complexity of the underlying Orlicz space. Moreover, asymptotically optimal codebooks induce a tight sequence of empirical measures. The set of possible accumulation points is characterized, and in most cases it consists of a single element. In that case, we find convergence as in the classical setting.